Speak Maths · Last school years

Sums and counting: stacking cans

Adding long lists, multiplying long lists and counting choices all have short signs. Once you can read them, a whole pattern fits on one line.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$a_{1}, a_{2}, \dots, a_{n}$ Say: a one, a two, and so on up to a nMeans: a row of numbers; the small number is the place, or index
$\sum_{i = 1}^{n} a_{i}$ Say: the sum of a i, for i from 1 to nMeans: add a 1, a 2, and so on up to a n
$\prod_{i = 1}^{n} a_{i}$ Say: the product of a i, for i from 1 to nMeans: multiply a 1, a 2, and so on up to a n
$n!$ Say: n factorialMeans: multiply every whole number from n down to 1
$(k n)$ Say: n choose kMeans: how many ways to pick k things from n, order not mattering
$\sum_{i = 1}^{n} i = \frac{n ( n + 1 )}{2}$ Say: the sum of i for i from 1 to n equals n times n plus 1, over 2Means: adding 1 up to n has a short formula

The story: Cans in a triangle

Subtitles:

Tap a line to see its words.

$a_{i} = i$ In a triangle of cans, row i has i cans.
$a_{1} + a_{2} + a_{3} = 6$ Three rows hold 1 + 2 + 3 = 6 cans.
$i = 1 \sum 4 i = 10$ Four rows hold 10 cans.
$i = 1 \sum n i = \frac{n ( n + 1 )}{2}$ Any number n of rows holds n times n plus 1, over 2.
$i = 1 \sum 10 i = \frac{10 \cdot 11}{2} = 55$ Ten rows need 55 cans.
$i = 1 \prod 4 i = 4! = 24$ Multiplying the four row sizes gives 24: that is 4 factorial.
$(2 5) = \frac{5 !}{2 ! 3 !} = 10$ Pick 2 of 5 cans for a display, and there are 10 ways.
$i = 1 \sum 4 i = (2 5)$ The same 10! Four rows hold as many cans as there are ways to pick 2 from 5.
$i = 1 \sum n i = (2 n + 1)$ It always works: n rows hold as many cans as there are ways to pick 2 from n + 1.

In words this story is 452 letters long; in symbols, 100.

Read it

What does it say?

$i = 1 \sum 3 i^{2}$

$5!$

$(2 6)$

$k = 1 \prod 4 2 = 16$

$k = 0 \sum 3 2^{k}$

$i = 1 \prod 3 2 i$

$(0 n) = 1$

$i = 1 \sum n 1 = n$

$(n + 1)! = (n + 1) \cdot n!$

$i = 1 \sum 20 i = 210$

$a_{i} = 3 i$

$(2 4) = 6$

Write it

Which notation says this?

Add the numbers from 1 to 100.

The number of ways to pick 3 things from 8.

Multiply all the whole numbers from 1 to n.

Add the squares of the numbers from 1 to 5.

Multiply the numbers from 1 to 6.

The number of ways to choose 2 things from 10.

The sum of the numbers from 1 to n, by the short formula.

The product of the first n even numbers.

The fifth term of the list a.

Add the odd numbers 1, 3, 5 and 7.

Three factorial times two factorial.

The sum of the numbers from 1 to 50 equals 50 times 51 over 2.

Build it

Tap the tiles in order. One tile does not belong.

The sum of i from 1 to 4 equals 10.

Five choose two equals five factorial over two factorial times three factorial.

Three factorial equals three times two times one.

The sum of i from 1 to n equals n plus 1 choose 2.

The product of i from 1 to 3 equals 6.

The list a one, a two, and so on up to a n.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 Lamps and nights

Four lamps, three nights. Let a_i be the number of nights lamp i was on. Judge each sentence: true or false?

	$N_{1}$	$N_{2}$	$N_{3}$
$L_{1}$	✓	✓	✓
$L_{2}$	✓	✓
$L_{3}$		✓	✓
$L_{4}$	✓

A tick means the lamp was on that night. a_i is the number of ticks in row i.

$\sum_{i = 1}^{4} a_{i} = 8$
$\sum_{i = 1}^{4} a_{i} = 9$
$a_{1} = a_{2} + a_{4}$
$\prod_{i = 1}^{3} a_{i} = 12$
$\sum_{i = 2}^{4} a_{i} = 6$
$\prod_{i = 1}^{4} a_{i} = 0$

2 Say it in words

Pair each line with what it asks you to do.

Tap a line, then its meaning.

3 A slip in the total

Adding 1 up to 10 should give the short formula's answer. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 Choosing five by two

This works out “5 choose 2”. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

5 Gauss in a minute

Add every whole number from 1 to 100. What do you get?

$i = 1 \sum 100 i$

6 A zero hides in the product

Here a_1 = 4, a_2 = 3, and so on. What is the product?

$a_{i} = 5 - i, i = 1 \prod 5 a_{i}$

7 Odd numbers add up

Add the first n odd numbers: 1, 1 + 3, 1 + 3 + 5, … you always get n squared (n = 4 gives 16). Write it in symbols, then compare.

Your version

8 Handshakes

n people each shake hands once with everyone else. Write the number of handshakes with a binomial, and as a formula. Then compare.

Your version

9 Cups of tea

Four days, three cups. Let a_i be the number of cups drunk on day i. Judge each sentence: true or false?

	$k_{1}$	$k_{2}$	$k_{3}$
$d_{1}$	✓	✓
$d_{2}$	✓	✓	✓
$d_{3}$			✓
$d_{4}$	✓		✓

A tick means that cup was drunk on that day; a_i counts the ticks in row i.

$\sum_{i = 1}^{4} a_{i} = 8$
$\prod_{i = 1}^{4} a_{i} = 12$
$a_{2} = \sum_{i = 3}^{4} a_{i}$
$\sum_{i = 1}^{2} a_{i} = \sum_{i = 3}^{4} a_{i}$
$\prod_{i = 1}^{4} a_{i} = \sum_{i = 1}^{4} a_{i}$
$a_{2}! = \sum_{i = 1}^{4} a_{i}$

10 The staircase

Cans on four steps: step i holds a_i cans, one per tick. Judge each sentence: true or false?

	$1$	$2$	$3$	$4$
$a_{1}$	✓
$a_{2}$	✓	✓
$a_{3}$	✓	✓	✓
$a_{4}$	✓	✓	✓	✓

A tick is one can; a_i is the number of ticks in row i, so a_i = i.

$\sum_{i = 1}^{4} a_{i} = \frac{4 \cdot 5}{2}$
$\prod_{i = 1}^{4} a_{i} = 4!$
$\sum_{i = 1}^{3} a_{i} = 2^{3}$
$\prod_{i = 1}^{4} a_{i} = \sum_{i = 1}^{4} a_{i}$
$\sum_{i = 1}^{4} a_{i} = (2 5)$
$a_{4}! = \sum_{i = 1}^{4} a_{i}$

11 Four cards

Four cards on the table. Pair each count with what it counts.

Tap a line, then its meaning.

12 Small print

Five lines that look alike. Pair each with the sum or product it stands for.

Tap a line, then its meaning.

13 The first four odd numbers

This should say: the first four odd numbers add up to 16. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

14 Six factorial

This works out 6! step by step, using 5! on the way. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

15 How many rows?

A triangle of cans has 21 cans in all. How many rows does it have?

$i = 1 \sum n i = 21 n = ?$

16 Shuffling the deck

n different cards can be laid in a row in n factorial ways, and n factorial is the product of all the numbers from 1 to n. Write it in symbols. Then compare.

Your version

Tell your own

Tell a three-line story about rows of something stacked or lined up, in signs only: add the rows with a sum, count the choices with n choose k. Then read it aloud.

Kept on this device only.

Sums and counting: stacking cans

The alphabet

The story: Cans in a triangle

Read it

Write it

Build it

Challenges

1 Lamps and nights

2 Say it in words

3 A slip in the total

4 Choosing five by two

5 Gauss in a minute

6 A zero hides in the product

7 Odd numbers add up

8 Handshakes

9 Cups of tea

10 The staircase

11 Four cards

12 Small print

13 The first four odd numbers

14 Six factorial

15 How many rows?

16 Shuffling the deck

Tell your own

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